LMFDB - Embedded newform 6039.2.a.j.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.546298$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.546298 q^{2} -1.70156 q^{4} +0.842631 q^{5} -4.19208 q^{7} -2.02216 q^{8} +O(q^{10})$	\(q+0.546298 q^{2} -1.70156 q^{4} +0.842631 q^{5} -4.19208 q^{7} -2.02216 q^{8} +0.460328 q^{10} +1.00000 q^{11} -5.95974 q^{13} -2.29012 q^{14} +2.29842 q^{16} -0.386200 q^{17} -2.95453 q^{19} -1.43379 q^{20} +0.546298 q^{22} +0.974992 q^{23} -4.28997 q^{25} -3.25580 q^{26} +7.13306 q^{28} +10.0563 q^{29} -10.6290 q^{31} +5.29993 q^{32} -0.210980 q^{34} -3.53237 q^{35} -3.16781 q^{37} -1.61405 q^{38} -1.70393 q^{40} +1.29317 q^{41} -4.99709 q^{43} -1.70156 q^{44} +0.532636 q^{46} -7.87272 q^{47} +10.5735 q^{49} -2.34361 q^{50} +10.1408 q^{52} -0.370002 q^{53} +0.842631 q^{55} +8.47703 q^{56} +5.49376 q^{58} -13.7723 q^{59} +1.00000 q^{61} -5.80661 q^{62} -1.70149 q^{64} -5.02186 q^{65} +9.80901 q^{67} +0.657141 q^{68} -1.92973 q^{70} +4.36706 q^{71} -2.06518 q^{73} -1.73057 q^{74} +5.02730 q^{76} -4.19208 q^{77} -8.67552 q^{79} +1.93672 q^{80} +0.706457 q^{82} +7.09673 q^{83} -0.325424 q^{85} -2.72990 q^{86} -2.02216 q^{88} +7.42432 q^{89} +24.9837 q^{91} -1.65901 q^{92} -4.30086 q^{94} -2.48957 q^{95} -6.00940 q^{97} +5.77628 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.546298	0.386291	0.193146	−	0.981170i	$-0.438131\pi$
$2$	0.546298	0.386291	0.193146	+	0.981170i	$0.438131\pi$
$3$	0	0
$3$	0	0
$4$	−1.70156	−0.850779
$5$	0.842631	0.376836	0.188418	−	0.982089i	$-0.439664\pi$
$5$	0.842631	0.376836	0.188418	+	0.982089i	$0.439664\pi$
$6$	0	0
$7$	−4.19208	−1.58446	−0.792228	−	0.610226i	$-0.791079\pi$
$7$	−4.19208	−1.58446	−0.792228	+	0.610226i	$0.791079\pi$
$8$	−2.02216	−0.714940
$9$	0	0
$10$	0.460328	0.145568
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.95974	−1.65293	−0.826467	−	0.562985i	$-0.809653\pi$
$13$	−5.95974	−1.65293	−0.826467	+	0.562985i	$0.809653\pi$
$14$	−2.29012	−0.612061
$15$	0	0
$16$	2.29842	0.574604
$17$	−0.386200	−0.0936672	−0.0468336	−	0.998903i	$-0.514913\pi$
$17$	−0.386200	−0.0936672	−0.0468336	+	0.998903i	$0.514913\pi$
$18$	0	0
$19$	−2.95453	−0.677815	−0.338907	−	0.940820i	$-0.610057\pi$
$19$	−2.95453	−0.677815	−0.338907	+	0.940820i	$0.610057\pi$
$20$	−1.43379	−0.320604
$21$	0	0
$22$	0.546298	0.116471
$23$	0.974992	0.203300	0.101650	−	0.994820i	$-0.467588\pi$
$23$	0.974992	0.203300	0.101650	+	0.994820i	$0.467588\pi$
$24$	0	0
$25$	−4.28997	−0.857995
$26$	−3.25580	−0.638514
$27$	0	0
$28$	7.13306	1.34802
$29$	10.0563	1.86741	0.933707	−	0.358038i	$-0.116554\pi$
$29$	10.0563	1.86741	0.933707	+	0.358038i	$0.116554\pi$
$30$	0	0
$31$	−10.6290	−1.90902	−0.954512	−	0.298171i	$-0.903623\pi$
$31$	−10.6290	−1.90902	−0.954512	+	0.298171i	$0.903623\pi$
$32$	5.29993	0.936904
$33$	0	0
$34$	−0.210980	−0.0361828
$35$	−3.53237	−0.597080
$36$	0	0
$37$	−3.16781	−0.520785	−0.260392	−	0.965503i	$-0.583852\pi$
$37$	−3.16781	−0.520785	−0.260392	+	0.965503i	$0.583852\pi$
$38$	−1.61405	−0.261834
$39$	0	0
$40$	−1.70393	−0.269415
$41$	1.29317	0.201959	0.100980	−	0.994888i	$-0.467802\pi$
$41$	1.29317	0.201959	0.100980	+	0.994888i	$0.467802\pi$
$42$	0	0
$43$	−4.99709	−0.762050	−0.381025	−	0.924565i	$-0.624429\pi$
$43$	−4.99709	−0.762050	−0.381025	+	0.924565i	$0.624429\pi$
$44$	−1.70156	−0.256520
$45$	0	0
$46$	0.532636	0.0785330
$47$	−7.87272	−1.14835	−0.574177	−	0.818731i	$-0.694678\pi$
$47$	−7.87272	−1.14835	−0.574177	+	0.818731i	$0.694678\pi$
$48$	0	0
$49$	10.5735	1.51050
$50$	−2.34361	−0.331436
$51$	0	0
$52$	10.1408	1.40628
$53$	−0.370002	−0.0508237	−0.0254119	−	0.999677i	$-0.508090\pi$
$53$	−0.370002	−0.0508237	−0.0254119	+	0.999677i	$0.508090\pi$
$54$	0	0
$55$	0.842631	0.113620
$56$	8.47703	1.13279
$57$	0	0
$58$	5.49376	0.721366
$59$	−13.7723	−1.79299	−0.896497	−	0.443049i	$-0.853897\pi$
$59$	−13.7723	−1.79299	−0.896497	+	0.443049i	$0.853897\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−5.80661	−0.737440
$63$	0	0
$64$	−1.70149	−0.212686
$65$	−5.02186	−0.622885
$66$	0	0
$67$	9.80901	1.19836	0.599181	−	0.800614i	$-0.295493\pi$
$67$	9.80901	1.19836	0.599181	+	0.800614i	$0.295493\pi$
$68$	0.657141	0.0796901
$69$	0	0
$70$	−1.92973	−0.230647
$71$	4.36706	0.518274	0.259137	−	0.965841i	$-0.416562\pi$
$71$	4.36706	0.518274	0.259137	+	0.965841i	$0.416562\pi$
$72$	0	0
$73$	−2.06518	−0.241711	−0.120855	−	0.992670i	$-0.538564\pi$
$73$	−2.06518	−0.241711	−0.120855	+	0.992670i	$0.538564\pi$
$74$	−1.73057	−0.201175
$75$	0	0
$76$	5.02730	0.576671
$77$	−4.19208	−0.477731
$78$	0	0
$79$	−8.67552	−0.976072	−0.488036	−	0.872823i	$-0.662287\pi$
$79$	−8.67552	−0.976072	−0.488036	+	0.872823i	$0.662287\pi$
$80$	1.93672	0.216531
$81$	0	0
$82$	0.706457	0.0780151
$83$	7.09673	0.778967	0.389483	−	0.921033i	$-0.372654\pi$
$83$	7.09673	0.778967	0.389483	+	0.921033i	$0.372654\pi$
$84$	0	0
$85$	−0.325424	−0.0352972
$86$	−2.72990	−0.294373
$87$	0	0
$88$	−2.02216	−0.215562
$89$	7.42432	0.786976	0.393488	−	0.919330i	$-0.371268\pi$
$89$	7.42432	0.786976	0.393488	+	0.919330i	$0.371268\pi$
$90$	0	0
$91$	24.9837	2.61900
$92$	−1.65901	−0.172963
$93$	0	0
$94$	−4.30086	−0.443599
$95$	−2.48957	−0.255425
$96$	0	0
$97$	−6.00940	−0.610162	−0.305081	−	0.952326i	$-0.598684\pi$
$97$	−6.00940	−0.610162	−0.305081	+	0.952326i	$0.598684\pi$
$98$	5.77628	0.583493
$99$	0	0
$100$	7.29964	0.729964
$101$	18.1851	1.80948	0.904742	−	0.425961i	$-0.140064\pi$
$101$	18.1851	1.80948	0.904742	+	0.425961i	$0.140064\pi$
$102$	0	0
$103$	10.9494	1.07887	0.539436	−	0.842026i	$-0.318637\pi$
$103$	10.9494	1.07887	0.539436	+	0.842026i	$0.318637\pi$
$104$	12.0515	1.18175
$105$	0	0
$106$	−0.202132	−0.0196328
$107$	−2.84318	−0.274860	−0.137430	−	0.990511i	$-0.543884\pi$
$107$	−2.84318	−0.274860	−0.137430	+	0.990511i	$0.543884\pi$
$108$	0	0
$109$	19.4494	1.86291	0.931456	−	0.363855i	$-0.118540\pi$
$109$	19.4494	1.86291	0.931456	+	0.363855i	$0.118540\pi$
$110$	0.460328	0.0438905
$111$	0	0
$112$	−9.63513	−0.910434
$113$	−8.08952	−0.760998	−0.380499	−	0.924781i	$-0.624248\pi$
$113$	−8.08952	−0.760998	−0.380499	+	0.924781i	$0.624248\pi$
$114$	0	0
$115$	0.821558	0.0766107
$116$	−17.1114	−1.58876
$117$	0	0
$118$	−7.52376	−0.692618
$119$	1.61898	0.148411
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.546298	0.0494595
$123$	0	0
$124$	18.0859	1.62416
$125$	−7.82802	−0.700159
$126$	0	0
$127$	1.18990	0.105587	0.0527934	−	0.998605i	$-0.483188\pi$
$127$	1.18990	0.105587	0.0527934	+	0.998605i	$0.483188\pi$
$128$	−11.5294	−1.01906
$129$	0	0
$130$	−2.74343	−0.240615
$131$	0.232276	0.0202940	0.0101470	−	0.999949i	$-0.496770\pi$
$131$	0.232276	0.0202940	0.0101470	+	0.999949i	$0.496770\pi$
$132$	0	0
$133$	12.3856	1.07397
$134$	5.35865	0.462917
$135$	0	0
$136$	0.780956	0.0669664
$137$	13.7946	1.17855	0.589276	−	0.807932i	$-0.299413\pi$
$137$	13.7946	1.17855	0.589276	+	0.807932i	$0.299413\pi$
$138$	0	0
$139$	10.2427	0.868777	0.434388	−	0.900726i	$-0.356965\pi$
$139$	10.2427	0.868777	0.434388	+	0.900726i	$0.356965\pi$
$140$	6.01053	0.507983
$141$	0	0
$142$	2.38572	0.200205
$143$	−5.95974	−0.498378
$144$	0	0
$145$	8.47377	0.703709
$146$	−1.12820	−0.0933708
$147$	0	0
$148$	5.39021	0.443073
$149$	2.70615	0.221697	0.110848	−	0.993837i	$-0.464643\pi$
$149$	2.70615	0.221697	0.110848	+	0.993837i	$0.464643\pi$
$150$	0	0
$151$	−15.6416	−1.27290	−0.636448	−	0.771319i	$-0.719597\pi$
$151$	−15.6416	−1.27290	−0.636448	+	0.771319i	$0.719597\pi$
$152$	5.97451	0.484597
$153$	0	0
$154$	−2.29012	−0.184543
$155$	−8.95632	−0.719389
$156$	0	0
$157$	3.50083	0.279397	0.139698	−	0.990194i	$-0.455387\pi$
$157$	3.50083	0.279397	0.139698	+	0.990194i	$0.455387\pi$
$158$	−4.73942	−0.377048
$159$	0	0
$160$	4.46589	0.353059
$161$	−4.08724	−0.322120
$162$	0	0
$163$	16.8255	1.31788	0.658938	−	0.752197i	$-0.271006\pi$
$163$	16.8255	1.31788	0.658938	+	0.752197i	$0.271006\pi$
$164$	−2.20040	−0.171823
$165$	0	0
$166$	3.87693	0.300908
$167$	−18.1086	−1.40129	−0.700645	−	0.713510i	$-0.747104\pi$
$167$	−18.1086	−1.40129	−0.700645	+	0.713510i	$0.747104\pi$
$168$	0	0
$169$	22.5185	1.73219
$170$	−0.177778	−0.0136350
$171$	0	0
$172$	8.50285	0.648336
$173$	9.84724	0.748672	0.374336	−	0.927293i	$-0.377871\pi$
$173$	9.84724	0.748672	0.374336	+	0.927293i	$0.377871\pi$
$174$	0	0
$175$	17.9839	1.35945
$176$	2.29842	0.173250
$177$	0	0
$178$	4.05589	0.304002
$179$	10.0310	0.749749	0.374875	−	0.927075i	$-0.377686\pi$
$179$	10.0310	0.749749	0.374875	+	0.927075i	$0.377686\pi$
$180$	0	0
$181$	−13.2298	−0.983362	−0.491681	−	0.870775i	$-0.663617\pi$
$181$	−13.2298	−0.983362	−0.491681	+	0.870775i	$0.663617\pi$
$182$	13.6485	1.01170
$183$	0	0
$184$	−1.97158	−0.145347
$185$	−2.66929	−0.196250
$186$	0	0
$187$	−0.386200	−0.0282417
$188$	13.3959	0.976996
$189$	0	0
$190$	−1.36005	−0.0986684
$191$	26.0670	1.88614	0.943069	−	0.332596i	$-0.107925\pi$
$191$	26.0670	1.88614	0.943069	+	0.332596i	$0.107925\pi$
$192$	0	0
$193$	−1.64976	−0.118752	−0.0593760	−	0.998236i	$-0.518911\pi$
$193$	−1.64976	−0.118752	−0.0593760	+	0.998236i	$0.518911\pi$
$194$	−3.28292	−0.235700
$195$	0	0
$196$	−17.9914	−1.28510
$197$	15.0092	1.06936	0.534681	−	0.845054i	$-0.320432\pi$
$197$	15.0092	1.06936	0.534681	+	0.845054i	$0.320432\pi$
$198$	0	0
$199$	0.432727	0.0306752	0.0153376	−	0.999882i	$-0.495118\pi$
$199$	0.432727	0.0306752	0.0153376	+	0.999882i	$0.495118\pi$
$200$	8.67499	0.613415
$201$	0	0
$202$	9.93448	0.698988
$203$	−42.1569	−2.95883
$204$	0	0
$205$	1.08967	0.0761055
$206$	5.98162	0.416759
$207$	0	0
$208$	−13.6980	−0.949783
$209$	−2.95453	−0.204369
$210$	0	0
$211$	−5.99457	−0.412683	−0.206342	−	0.978480i	$-0.566156\pi$
$211$	−5.99457	−0.412683	−0.206342	+	0.978480i	$0.566156\pi$
$212$	0.629580	0.0432398
$213$	0	0
$214$	−1.55322	−0.106176
$215$	−4.21070	−0.287168
$216$	0	0
$217$	44.5576	3.02476
$218$	10.6252	0.719626
$219$	0	0
$220$	−1.43379	−0.0966658
$221$	2.30165	0.154826
$222$	0	0
$223$	1.20744	0.0808559	0.0404280	−	0.999182i	$-0.487128\pi$
$223$	1.20744	0.0808559	0.0404280	+	0.999182i	$0.487128\pi$
$224$	−22.2177	−1.48448
$225$	0	0
$226$	−4.41929	−0.293967
$227$	−3.17333	−0.210621	−0.105311	−	0.994439i	$-0.533584\pi$
$227$	−3.17333	−0.210621	−0.105311	+	0.994439i	$0.533584\pi$
$228$	0	0
$229$	−26.1824	−1.73018	−0.865091	−	0.501614i	$-0.832740\pi$
$229$	−26.1824	−1.73018	−0.865091	+	0.501614i	$0.832740\pi$
$230$	0.448816	0.0295940
$231$	0	0
$232$	−20.3355	−1.33509
$233$	−15.3586	−1.00617	−0.503087	−	0.864236i	$-0.667803\pi$
$233$	−15.3586	−1.00617	−0.503087	+	0.864236i	$0.667803\pi$
$234$	0	0
$235$	−6.63380	−0.432741
$236$	23.4343	1.52544
$237$	0	0
$238$	0.884445	0.0573301
$239$	6.48318	0.419362	0.209681	−	0.977770i	$-0.432757\pi$
$239$	6.48318	0.419362	0.209681	+	0.977770i	$0.432757\pi$
$240$	0	0
$241$	−2.05017	−0.132063	−0.0660315	−	0.997818i	$-0.521034\pi$
$241$	−2.05017	−0.132063	−0.0660315	+	0.997818i	$0.521034\pi$
$242$	0.546298	0.0351174
$243$	0	0
$244$	−1.70156	−0.108931
$245$	8.90955	0.569210
$246$	0	0
$247$	17.6082	1.12038
$248$	21.4935	1.36484
$249$	0	0
$250$	−4.27643	−0.270465
$251$	−0.403663	−0.0254790	−0.0127395	−	0.999919i	$-0.504055\pi$
$251$	−0.403663	−0.0254790	−0.0127395	+	0.999919i	$0.504055\pi$
$252$	0	0
$253$	0.974992	0.0612972
$254$	0.650042	0.0407873
$255$	0	0
$256$	−2.89551	−0.180969
$257$	3.52100	0.219634	0.109817	−	0.993952i	$-0.464973\pi$
$257$	3.52100	0.219634	0.109817	+	0.993952i	$0.464973\pi$
$258$	0	0
$259$	13.2797	0.825160
$260$	8.54498	0.529937
$261$	0	0
$262$	0.126892	0.00783940
$263$	−25.3122	−1.56082	−0.780408	−	0.625271i	$-0.784988\pi$
$263$	−25.3122	−1.56082	−0.780408	+	0.625271i	$0.784988\pi$
$264$	0	0
$265$	−0.311775	−0.0191522
$266$	6.76623	0.414864
$267$	0	0
$268$	−16.6906	−1.01954
$269$	−26.8357	−1.63620	−0.818100	−	0.575076i	$-0.804972\pi$
$269$	−26.8357	−1.63620	−0.818100	+	0.575076i	$0.804972\pi$
$270$	0	0
$271$	−8.80885	−0.535099	−0.267550	−	0.963544i	$-0.586214\pi$
$271$	−8.80885	−0.535099	−0.267550	+	0.963544i	$0.586214\pi$
$272$	−0.887648	−0.0538215
$273$	0	0
$274$	7.53597	0.455264
$275$	−4.28997	−0.258695
$276$	0	0
$277$	21.3581	1.28328	0.641641	−	0.767005i	$-0.278254\pi$
$277$	21.3581	1.28328	0.641641	+	0.767005i	$0.278254\pi$
$278$	5.59559	0.335601
$279$	0	0
$280$	7.14300	0.426876
$281$	29.9872	1.78889	0.894444	−	0.447181i	$-0.147572\pi$
$281$	29.9872	1.78889	0.894444	+	0.447181i	$0.147572\pi$
$282$	0	0
$283$	10.0912	0.599862	0.299931	−	0.953961i	$-0.403036\pi$
$283$	10.0912	0.599862	0.299931	+	0.953961i	$0.403036\pi$
$284$	−7.43080	−0.440937
$285$	0	0
$286$	−3.25580	−0.192519
$287$	−5.42107	−0.319996
$288$	0	0
$289$	−16.8508	−0.991226
$290$	4.62921	0.271837
$291$	0	0
$292$	3.51402	0.205643
$293$	11.7387	0.685783	0.342892	−	0.939375i	$-0.388594\pi$
$293$	11.7387	0.685783	0.342892	+	0.939375i	$0.388594\pi$
$294$	0	0
$295$	−11.6049	−0.675665
$296$	6.40580	0.372330
$297$	0	0
$298$	1.47837	0.0856395
$299$	−5.81070	−0.336041
$300$	0	0
$301$	20.9482	1.20743
$302$	−8.54499	−0.491709
$303$	0	0
$304$	−6.79073	−0.389475
$305$	0.842631	0.0482489
$306$	0	0
$307$	12.1646	0.694269	0.347135	−	0.937815i	$-0.387155\pi$
$307$	12.1646	0.694269	0.347135	+	0.937815i	$0.387155\pi$
$308$	7.13306	0.406444
$309$	0	0
$310$	−4.89282	−0.277894
$311$	10.4238	0.591079	0.295539	−	0.955331i	$-0.404501\pi$
$311$	10.4238	0.591079	0.295539	+	0.955331i	$0.404501\pi$
$312$	0	0
$313$	−3.70708	−0.209536	−0.104768	−	0.994497i	$-0.533410\pi$
$313$	−3.70708	−0.209536	−0.104768	+	0.994497i	$0.533410\pi$
$314$	1.91250	0.107928
$315$	0	0
$316$	14.7619	0.830422
$317$	17.7635	0.997696	0.498848	−	0.866690i	$-0.333757\pi$
$317$	17.7635	0.997696	0.498848	+	0.866690i	$0.333757\pi$
$318$	0	0
$319$	10.0563	0.563046
$320$	−1.43373	−0.0801477
$321$	0	0
$322$	−2.23285	−0.124432
$323$	1.14104	0.0634890
$324$	0	0
$325$	25.5671	1.41821
$326$	9.19176	0.509084
$327$	0	0
$328$	−2.61499	−0.144389
$329$	33.0030	1.81952
$330$	0	0
$331$	25.2813	1.38959	0.694794	−	0.719209i	$-0.255496\pi$
$331$	25.2813	1.38959	0.694794	+	0.719209i	$0.255496\pi$
$332$	−12.0755	−0.662729
$333$	0	0
$334$	−9.89273	−0.541306
$335$	8.26538	0.451586
$336$	0	0
$337$	−15.8267	−0.862134	−0.431067	−	0.902320i	$-0.641863\pi$
$337$	−15.8267	−0.862134	−0.431067	+	0.902320i	$0.641863\pi$
$338$	12.3018	0.669130
$339$	0	0
$340$	0.553727	0.0300301
$341$	−10.6290	−0.575593
$342$	0	0
$343$	−14.9804	−0.808863
$344$	10.1049	0.544820
$345$	0	0
$346$	5.37953	0.289205
$347$	14.2210	0.763424	0.381712	−	0.924281i	$-0.375335\pi$
$347$	14.2210	0.763424	0.381712	+	0.924281i	$0.375335\pi$
$348$	0	0
$349$	20.6858	1.10728	0.553642	−	0.832755i	$-0.313238\pi$
$349$	20.6858	1.10728	0.553642	+	0.832755i	$0.313238\pi$
$350$	9.82457	0.525145
$351$	0	0
$352$	5.29993	0.282487
$353$	13.3604	0.711102	0.355551	−	0.934657i	$-0.384293\pi$
$353$	13.3604	0.711102	0.355551	+	0.934657i	$0.384293\pi$
$354$	0	0
$355$	3.67981	0.195304
$356$	−12.6329	−0.669543
$357$	0	0
$358$	5.47990	0.289622
$359$	10.9660	0.578761	0.289381	−	0.957214i	$-0.406551\pi$
$359$	10.9660	0.578761	0.289381	+	0.957214i	$0.406551\pi$
$360$	0	0
$361$	−10.2708	−0.540567
$362$	−7.22741	−0.379864
$363$	0	0
$364$	−42.5112	−2.22819
$365$	−1.74018	−0.0910854
$366$	0	0
$367$	−18.2275	−0.951468	−0.475734	−	0.879589i	$-0.657818\pi$
$367$	−18.2275	−0.951468	−0.475734	+	0.879589i	$0.657818\pi$
$368$	2.24094	0.116817
$369$	0	0
$370$	−1.45823	−0.0758098
$371$	1.55108	0.0805279
$372$	0	0
$373$	−14.3974	−0.745467	−0.372733	−	0.927938i	$-0.621579\pi$
$373$	−14.3974	−0.745467	−0.372733	+	0.927938i	$0.621579\pi$
$374$	−0.210980	−0.0109095
$375$	0	0
$376$	15.9199	0.821005
$377$	−59.9331	−3.08671
$378$	0	0
$379$	11.1418	0.572316	0.286158	−	0.958182i	$-0.407622\pi$
$379$	11.1418	0.572316	0.286158	+	0.958182i	$0.407622\pi$
$380$	4.23616	0.217310
$381$	0	0
$382$	14.2403	0.728599
$383$	13.6433	0.697140	0.348570	−	0.937283i	$-0.386667\pi$
$383$	13.6433	0.697140	0.348570	+	0.937283i	$0.386667\pi$
$384$	0	0
$385$	−3.53237	−0.180026
$386$	−0.901260	−0.0458729
$387$	0	0
$388$	10.2253	0.519113
$389$	−23.6998	−1.20163	−0.600814	−	0.799389i	$-0.705157\pi$
$389$	−23.6998	−1.20163	−0.600814	+	0.799389i	$0.705157\pi$
$390$	0	0
$391$	−0.376542	−0.0190425
$392$	−21.3812	−1.07992
$393$	0	0
$394$	8.19951	0.413086
$395$	−7.31026	−0.367819
$396$	0	0
$397$	−19.8855	−0.998023	−0.499011	−	0.866595i	$-0.666303\pi$
$397$	−19.8855	−0.998023	−0.499011	+	0.866595i	$0.666303\pi$
$398$	0.236398	0.0118496
$399$	0	0
$400$	−9.86014	−0.493007
$401$	15.1693	0.757517	0.378758	−	0.925496i	$-0.376351\pi$
$401$	15.1693	0.757517	0.378758	+	0.925496i	$0.376351\pi$
$402$	0	0
$403$	63.3461	3.15549
$404$	−30.9430	−1.53947
$405$	0	0
$406$	−23.0302	−1.14297
$407$	−3.16781	−0.157022
$408$	0	0
$409$	−20.3853	−1.00799	−0.503995	−	0.863706i	$-0.668137\pi$
$409$	−20.3853	−1.00799	−0.503995	+	0.863706i	$0.668137\pi$
$410$	0.595282	0.0293989
$411$	0	0
$412$	−18.6310	−0.917882
$413$	57.7343	2.84092
$414$	0	0
$415$	5.97992	0.293543
$416$	−31.5862	−1.54864
$417$	0	0
$418$	−1.61405	−0.0789459
$419$	−19.1519	−0.935633	−0.467817	−	0.883826i	$-0.654959\pi$
$419$	−19.1519	−0.935633	−0.467817	+	0.883826i	$0.654959\pi$
$420$	0	0
$421$	−8.29681	−0.404362	−0.202181	−	0.979348i	$-0.564803\pi$
$421$	−8.29681	−0.404362	−0.202181	+	0.979348i	$0.564803\pi$
$422$	−3.27482	−0.159416
$423$	0	0
$424$	0.748202	0.0363359
$425$	1.65679	0.0803660
$426$	0	0
$427$	−4.19208	−0.202869
$428$	4.83783	0.233845
$429$	0	0
$430$	−2.30030	−0.110930
$431$	17.6981	0.852485	0.426243	−	0.904609i	$-0.359837\pi$
$431$	17.6981	0.852485	0.426243	+	0.904609i	$0.359837\pi$
$432$	0	0
$433$	5.57917	0.268118	0.134059	−	0.990973i	$-0.457199\pi$
$433$	5.57917	0.268118	0.134059	+	0.990973i	$0.457199\pi$
$434$	24.3417	1.16844
$435$	0	0
$436$	−33.0942	−1.58493
$437$	−2.88064	−0.137800
$438$	0	0
$439$	−6.90623	−0.329616	−0.164808	−	0.986326i	$-0.552701\pi$
$439$	−6.90623	−0.329616	−0.164808	+	0.986326i	$0.552701\pi$
$440$	−1.70393	−0.0812317
$441$	0	0
$442$	1.25739	0.0598078
$443$	−8.54914	−0.406182	−0.203091	−	0.979160i	$-0.565099\pi$
$443$	−8.54914	−0.406182	−0.203091	+	0.979160i	$0.565099\pi$
$444$	0	0
$445$	6.25596	0.296561
$446$	0.659621	0.0312339
$447$	0	0
$448$	7.13276	0.336991
$449$	22.1028	1.04309	0.521547	−	0.853223i	$-0.325355\pi$
$449$	22.1028	1.04309	0.521547	+	0.853223i	$0.325355\pi$
$450$	0	0
$451$	1.29317	0.0608930
$452$	13.7648	0.647441
$453$	0	0
$454$	−1.73358	−0.0813611
$455$	21.0520	0.986933
$456$	0	0
$457$	−12.2492	−0.572995	−0.286497	−	0.958081i	$-0.592491\pi$
$457$	−12.2492	−0.572995	−0.286497	+	0.958081i	$0.592491\pi$
$458$	−14.3034	−0.668355
$459$	0	0
$460$	−1.39793	−0.0651788
$461$	5.51693	0.256949	0.128475	−	0.991713i	$-0.458992\pi$
$461$	5.51693	0.256949	0.128475	+	0.991713i	$0.458992\pi$
$462$	0	0
$463$	−39.0620	−1.81536	−0.907682	−	0.419659i	$-0.862150\pi$
$463$	−39.0620	−1.81536	−0.907682	+	0.419659i	$0.862150\pi$
$464$	23.1136	1.07302
$465$	0	0
$466$	−8.39037	−0.388676
$467$	15.4534	0.715099	0.357550	−	0.933894i	$-0.383612\pi$
$467$	15.4534	0.715099	0.357550	+	0.933894i	$0.383612\pi$
$468$	0	0
$469$	−41.1201	−1.89875
$470$	−3.62403	−0.167164
$471$	0	0
$472$	27.8496	1.28188
$473$	−4.99709	−0.229767
$474$	0	0
$475$	12.6748	0.581561
$476$	−2.75479	−0.126265
$477$	0	0
$478$	3.54175	0.161996
$479$	−19.2183	−0.878106	−0.439053	−	0.898461i	$-0.644686\pi$
$479$	−19.2183	−0.878106	−0.439053	+	0.898461i	$0.644686\pi$
$480$	0	0
$481$	18.8793	0.860823
$482$	−1.12000	−0.0510148
$483$	0	0
$484$	−1.70156	−0.0773435
$485$	−5.06370	−0.229931
$486$	0	0
$487$	−5.36667	−0.243187	−0.121594	−	0.992580i	$-0.538800\pi$
$487$	−5.36667	−0.243187	−0.121594	+	0.992580i	$0.538800\pi$
$488$	−2.02216	−0.0915387
$489$	0	0
$490$	4.86727	0.219881
$491$	30.3986	1.37187	0.685936	−	0.727662i	$-0.259393\pi$
$491$	30.3986	1.37187	0.685936	+	0.727662i	$0.259393\pi$
$492$	0	0
$493$	−3.88375	−0.174915
$494$	9.61933	0.432794
$495$	0	0
$496$	−24.4299	−1.09693
$497$	−18.3070	−0.821182
$498$	0	0
$499$	−17.0924	−0.765160	−0.382580	−	0.923922i	$-0.624964\pi$
$499$	−17.0924	−0.765160	−0.382580	+	0.923922i	$0.624964\pi$
$500$	13.3198	0.595681
$501$	0	0
$502$	−0.220521	−0.00984231
$503$	−32.1659	−1.43421	−0.717103	−	0.696967i	$-0.754532\pi$
$503$	−32.1659	−1.43421	−0.717103	+	0.696967i	$0.754532\pi$
$504$	0	0
$505$	15.3233	0.681878
$506$	0.532636	0.0236786
$507$	0	0
$508$	−2.02469	−0.0898310
$509$	−20.2801	−0.898900	−0.449450	−	0.893306i	$-0.648380\pi$
$509$	−20.2801	−0.898900	−0.449450	+	0.893306i	$0.648380\pi$
$510$	0	0
$511$	8.65739	0.382980
$512$	21.4770	0.949156
$513$	0	0
$514$	1.92352	0.0848428
$515$	9.22627	0.406558
$516$	0	0
$517$	−7.87272	−0.346242
$518$	7.25467	0.318752
$519$	0	0
$520$	10.1550	0.445325
$521$	−27.3063	−1.19631	−0.598156	−	0.801380i	$-0.704100\pi$
$521$	−27.3063	−1.19631	−0.598156	+	0.801380i	$0.704100\pi$
$522$	0	0
$523$	−23.5138	−1.02819	−0.514093	−	0.857734i	$-0.671871\pi$
$523$	−23.5138	−1.02819	−0.514093	+	0.857734i	$0.671871\pi$
$524$	−0.395230	−0.0172657
$525$	0	0
$526$	−13.8280	−0.602929
$527$	4.10492	0.178813
$528$	0	0
$529$	−22.0494	−0.958669
$530$	−0.170322	−0.00739833
$531$	0	0
$532$	−21.0748	−0.913709
$533$	−7.70696	−0.333825
$534$	0	0
$535$	−2.39575	−0.103577
$536$	−19.8353	−0.856757
$537$	0	0
$538$	−14.6603	−0.632050
$539$	10.5735	0.455433
$540$	0	0
$541$	21.3883	0.919556	0.459778	−	0.888034i	$-0.347929\pi$
$541$	21.3883	0.919556	0.459778	+	0.888034i	$0.347929\pi$
$542$	−4.81226	−0.206704
$543$	0	0
$544$	−2.04683	−0.0877572
$545$	16.3886	0.702012
$546$	0	0
$547$	−18.4841	−0.790322	−0.395161	−	0.918612i	$-0.629311\pi$
$547$	−18.4841	−0.790322	−0.395161	+	0.918612i	$0.629311\pi$
$548$	−23.4723	−1.00269
$549$	0	0
$550$	−2.34361	−0.0999317
$551$	−29.7117	−1.26576
$552$	0	0
$553$	36.3684	1.54654
$554$	11.6679	0.495721
$555$	0	0
$556$	−17.4286	−0.739137
$557$	24.4886	1.03762	0.518808	−	0.854891i	$-0.326376\pi$
$557$	24.4886	1.03762	0.518808	+	0.854891i	$0.326376\pi$
$558$	0	0
$559$	29.7814	1.25962
$560$	−8.11886	−0.343084
$561$	0	0
$562$	16.3820	0.691032
$563$	20.3211	0.856433	0.428216	−	0.903676i	$-0.359142\pi$
$563$	20.3211	0.856433	0.428216	+	0.903676i	$0.359142\pi$
$564$	0	0
$565$	−6.81648	−0.286771
$566$	5.51283	0.231721
$567$	0	0
$568$	−8.83086	−0.370535
$569$	24.7529	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$569$	24.7529	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$570$	0	0
$571$	10.7099	0.448195	0.224098	−	0.974567i	$-0.428057\pi$
$571$	10.7099	0.448195	0.224098	+	0.974567i	$0.428057\pi$
$572$	10.1408	0.424010
$573$	0	0
$574$	−2.96152	−0.123612
$575$	−4.18269	−0.174430
$576$	0	0
$577$	−4.97286	−0.207023	−0.103512	−	0.994628i	$-0.533008\pi$
$577$	−4.97286	−0.207023	−0.103512	+	0.994628i	$0.533008\pi$
$578$	−9.20559	−0.382902
$579$	0	0
$580$	−14.4186	−0.598701
$581$	−29.7500	−1.23424
$582$	0	0
$583$	−0.370002	−0.0153239
$584$	4.17611	0.172809
$585$	0	0
$586$	6.41284	0.264912
$587$	−2.13571	−0.0881502	−0.0440751	−	0.999028i	$-0.514034\pi$
$587$	−2.13571	−0.0881502	−0.0440751	+	0.999028i	$0.514034\pi$
$588$	0	0
$589$	31.4037	1.29397
$590$	−6.33975	−0.261003
$591$	0	0
$592$	−7.28094	−0.299245
$593$	−13.9935	−0.574646	−0.287323	−	0.957834i	$-0.592765\pi$
$593$	−13.9935	−0.574646	−0.287323	+	0.957834i	$0.592765\pi$
$594$	0	0
$595$	1.36420	0.0559268
$596$	−4.60468	−0.188615
$597$	0	0
$598$	−3.17437	−0.129810
$599$	27.2804	1.11465	0.557324	−	0.830295i	$-0.311828\pi$
$599$	27.2804	1.11465	0.557324	+	0.830295i	$0.311828\pi$
$600$	0	0
$601$	38.7081	1.57894	0.789468	−	0.613791i	$-0.210356\pi$
$601$	38.7081	1.57894	0.789468	+	0.613791i	$0.210356\pi$
$602$	11.4440	0.466421
$603$	0	0
$604$	26.6151	1.08295
$605$	0.842631	0.0342578
$606$	0	0
$607$	27.3959	1.11197	0.555983	−	0.831194i	$-0.312342\pi$
$607$	27.3959	1.11197	0.555983	+	0.831194i	$0.312342\pi$
$608$	−15.6588	−0.635048
$609$	0	0
$610$	0.460328	0.0186381
$611$	46.9194	1.89815
$612$	0	0
$613$	8.56875	0.346088	0.173044	−	0.984914i	$-0.444640\pi$
$613$	8.56875	0.346088	0.173044	+	0.984914i	$0.444640\pi$
$614$	6.64549	0.268190
$615$	0	0
$616$	8.47703	0.341549
$617$	−45.6108	−1.83622	−0.918112	−	0.396322i	$-0.870287\pi$
$617$	−45.6108	−1.83622	−0.918112	+	0.396322i	$0.870287\pi$
$618$	0	0
$619$	36.2410	1.45665	0.728325	−	0.685232i	$-0.240299\pi$
$619$	36.2410	1.45665	0.728325	+	0.685232i	$0.240299\pi$
$620$	15.2397	0.612041
$621$	0	0
$622$	5.69450	0.228329
$623$	−31.1233	−1.24693
$624$	0	0
$625$	14.8537	0.594150
$626$	−2.02517	−0.0809421
$627$	0	0
$628$	−5.95686	−0.237705
$629$	1.22341	0.0487804
$630$	0	0
$631$	−5.90488	−0.235070	−0.117535	−	0.993069i	$-0.537499\pi$
$631$	−5.90488	−0.235070	−0.117535	+	0.993069i	$0.537499\pi$
$632$	17.5432	0.697833
$633$	0	0
$634$	9.70415	0.385401
$635$	1.00265	0.0397889
$636$	0	0
$637$	−63.0153	−2.49676
$638$	5.49376	0.217500
$639$	0	0
$640$	−9.71501	−0.384020
$641$	27.6940	1.09385	0.546923	−	0.837183i	$-0.315799\pi$
$641$	27.6940	1.09385	0.546923	+	0.837183i	$0.315799\pi$
$642$	0	0
$643$	39.7962	1.56941	0.784704	−	0.619871i	$-0.212815\pi$
$643$	39.7962	1.56941	0.784704	+	0.619871i	$0.212815\pi$
$644$	6.95467	0.274053
$645$	0	0
$646$	0.623347	0.0245253
$647$	−35.8326	−1.40872	−0.704362	−	0.709841i	$-0.748767\pi$
$647$	−35.8326	−1.40872	−0.704362	+	0.709841i	$0.748767\pi$
$648$	0	0
$649$	−13.7723	−0.540608
$650$	13.9673	0.547842
$651$	0	0
$652$	−28.6296	−1.12122
$653$	−3.93565	−0.154014	−0.0770070	−	0.997031i	$-0.524536\pi$
$653$	−3.93565	−0.154014	−0.0770070	+	0.997031i	$0.524536\pi$
$654$	0	0
$655$	0.195723	0.00764751
$656$	2.97224	0.116047
$657$	0	0
$658$	18.0295	0.702864
$659$	1.14603	0.0446429	0.0223215	−	0.999751i	$-0.492894\pi$
$659$	1.14603	0.0446429	0.0223215	+	0.999751i	$0.492894\pi$
$660$	0	0
$661$	−21.1185	−0.821414	−0.410707	−	0.911767i	$-0.634718\pi$
$661$	−21.1185	−0.821414	−0.410707	+	0.911767i	$0.634718\pi$
$662$	13.8112	0.536786
$663$	0	0
$664$	−14.3507	−0.556915
$665$	10.4365	0.404709
$666$	0	0
$667$	9.80484	0.379645
$668$	30.8129	1.19219
$669$	0	0
$670$	4.51536	0.174444
$671$	1.00000	0.0386046
$672$	0	0
$673$	−50.2053	−1.93527	−0.967636	−	0.252349i	$-0.918797\pi$
$673$	−50.2053	−1.93527	−0.967636	+	0.252349i	$0.918797\pi$
$674$	−8.64609	−0.333035
$675$	0	0
$676$	−38.3165	−1.47371
$677$	15.0864	0.579819	0.289910	−	0.957054i	$-0.406375\pi$
$677$	15.0864	0.579819	0.289910	+	0.957054i	$0.406375\pi$
$678$	0	0
$679$	25.1918	0.966774
$680$	0.658057	0.0252353
$681$	0	0
$682$	−5.80661	−0.222346
$683$	−48.5108	−1.85621	−0.928107	−	0.372314i	$-0.878564\pi$
$683$	−48.5108	−1.85621	−0.928107	+	0.372314i	$0.878564\pi$
$684$	0	0
$685$	11.6238	0.444121
$686$	−8.18375	−0.312457
$687$	0	0
$688$	−11.4854	−0.437877
$689$	2.20512	0.0840083
$690$	0	0
$691$	−28.3558	−1.07871	−0.539353	−	0.842080i	$-0.681331\pi$
$691$	−28.3558	−1.07871	−0.539353	+	0.842080i	$0.681331\pi$
$692$	−16.7557	−0.636954
$693$	0	0
$694$	7.76891	0.294904
$695$	8.63084	0.327386
$696$	0	0
$697$	−0.499422	−0.0189170
$698$	11.3006	0.427734
$699$	0	0
$700$	−30.6006	−1.15660
$701$	4.65092	0.175663	0.0878313	−	0.996135i	$-0.472006\pi$
$701$	4.65092	0.175663	0.0878313	+	0.996135i	$0.472006\pi$
$702$	0	0
$703$	9.35937	0.352995
$704$	−1.70149	−0.0641272
$705$	0	0
$706$	7.29876	0.274692
$707$	−76.2332	−2.86705
$708$	0	0
$709$	47.0222	1.76595	0.882977	−	0.469416i	$-0.155536\pi$
$709$	47.0222	1.76595	0.882977	+	0.469416i	$0.155536\pi$
$710$	2.01028	0.0754443
$711$	0	0
$712$	−15.0131	−0.562641
$713$	−10.3632	−0.388104
$714$	0	0
$715$	−5.02186	−0.187807
$716$	−17.0683	−0.637871
$717$	0	0
$718$	5.99069	0.223570
$719$	−8.60663	−0.320973	−0.160487	−	0.987038i	$-0.551306\pi$
$719$	−8.60663	−0.320973	−0.160487	+	0.987038i	$0.551306\pi$
$720$	0	0
$721$	−45.9005	−1.70943
$722$	−5.61091	−0.208816
$723$	0	0
$724$	22.5112	0.836624
$725$	−43.1414	−1.60223
$726$	0	0
$727$	29.4993	1.09407	0.547034	−	0.837110i	$-0.315757\pi$
$727$	29.4993	1.09407	0.547034	+	0.837110i	$0.315757\pi$
$728$	−50.5209	−1.87243
$729$	0	0
$730$	−0.950659	−0.0351855
$731$	1.92988	0.0713791
$732$	0	0
$733$	−42.0201	−1.55205	−0.776025	−	0.630702i	$-0.782767\pi$
$733$	−42.0201	−1.55205	−0.776025	+	0.630702i	$0.782767\pi$
$734$	−9.95766	−0.367544
$735$	0	0
$736$	5.16739	0.190473
$737$	9.80901	0.361320
$738$	0	0
$739$	50.8992	1.87236	0.936179	−	0.351523i	$-0.114336\pi$
$739$	50.8992	1.87236	0.936179	+	0.351523i	$0.114336\pi$
$740$	4.54196	0.166966
$741$	0	0
$742$	0.847351	0.0311072
$743$	4.44260	0.162983	0.0814916	−	0.996674i	$-0.474032\pi$
$743$	4.44260	0.162983	0.0814916	+	0.996674i	$0.474032\pi$
$744$	0	0
$745$	2.28029	0.0835433
$746$	−7.86525	−0.287967
$747$	0	0
$748$	0.657141	0.0240275
$749$	11.9188	0.435504
$750$	0	0
$751$	3.39662	0.123944	0.0619722	−	0.998078i	$-0.480261\pi$
$751$	3.39662	0.123944	0.0619722	+	0.998078i	$0.480261\pi$
$752$	−18.0948	−0.659849
$753$	0	0
$754$	−32.7414	−1.19237
$755$	−13.1801	−0.479673
$756$	0	0
$757$	13.8258	0.502508	0.251254	−	0.967921i	$-0.419157\pi$
$757$	13.8258	0.502508	0.251254	+	0.967921i	$0.419157\pi$
$758$	6.08675	0.221081
$759$	0	0
$760$	5.03431	0.182613
$761$	34.7597	1.26004	0.630018	−	0.776580i	$-0.283047\pi$
$761$	34.7597	1.26004	0.630018	+	0.776580i	$0.283047\pi$
$762$	0	0
$763$	−81.5332	−2.95170
$764$	−44.3544	−1.60469
$765$	0	0
$766$	7.45331	0.269299
$767$	82.0790	2.96370
$768$	0	0
$769$	−1.09955	−0.0396507	−0.0198254	−	0.999803i	$-0.506311\pi$
$769$	−1.09955	−0.0396507	−0.0198254	+	0.999803i	$0.506311\pi$
$770$	−1.92973	−0.0695426
$771$	0	0
$772$	2.80716	0.101032
$773$	13.8425	0.497879	0.248939	−	0.968519i	$-0.419918\pi$
$773$	13.8425	0.497879	0.248939	+	0.968519i	$0.419918\pi$
$774$	0	0
$775$	45.5981	1.63793
$776$	12.1519	0.436229
$777$	0	0
$778$	−12.9472	−0.464178
$779$	−3.82071	−0.136891
$780$	0	0
$781$	4.36706	0.156266
$782$	−0.205704	−0.00735596
$783$	0	0
$784$	24.3023	0.867939
$785$	2.94991	0.105287
$786$	0	0
$787$	−33.6766	−1.20044	−0.600220	−	0.799835i	$-0.704920\pi$
$787$	−33.6766	−1.20044	−0.600220	+	0.799835i	$0.704920\pi$
$788$	−25.5391	−0.909792
$789$	0	0
$790$	−3.99358	−0.142085
$791$	33.9119	1.20577
$792$	0	0
$793$	−5.95974	−0.211637
$794$	−10.8634	−0.385528
$795$	0	0
$796$	−0.736310	−0.0260978
$797$	1.47756	0.0523377	0.0261689	−	0.999658i	$-0.491669\pi$
$797$	1.47756	0.0523377	0.0261689	+	0.999658i	$0.491669\pi$
$798$	0	0
$799$	3.04044	0.107563
$800$	−22.7366	−0.803859
$801$	0	0
$802$	8.28694	0.292622
$803$	−2.06518	−0.0728786
$804$	0	0
$805$	−3.44403	−0.121386
$806$	34.6059	1.21894
$807$	0	0
$808$	−36.7731	−1.29367
$809$	−26.0611	−0.916260	−0.458130	−	0.888885i	$-0.651481\pi$
$809$	−26.0611	−0.916260	−0.458130	+	0.888885i	$0.651481\pi$
$810$	0	0
$811$	21.1286	0.741925	0.370962	−	0.928648i	$-0.379028\pi$
$811$	21.1286	0.741925	0.370962	+	0.928648i	$0.379028\pi$
$812$	71.7324	2.51731
$813$	0	0
$814$	−1.73057	−0.0606564
$815$	14.1777	0.496623
$816$	0	0
$817$	14.7640	0.516529
$818$	−11.1365	−0.389378
$819$	0	0
$820$	−1.85413	−0.0647490
$821$	46.0338	1.60659	0.803295	−	0.595582i	$-0.203078\pi$
$821$	46.0338	1.60659	0.803295	+	0.595582i	$0.203078\pi$
$822$	0	0
$823$	−23.0217	−0.802488	−0.401244	−	0.915971i	$-0.631422\pi$
$823$	−23.0217	−0.802488	−0.401244	+	0.915971i	$0.631422\pi$
$824$	−22.1413	−0.771329
$825$	0	0
$826$	31.5402	1.09742
$827$	15.9583	0.554923	0.277461	−	0.960737i	$-0.410507\pi$
$827$	15.9583	0.554923	0.277461	+	0.960737i	$0.410507\pi$
$828$	0	0
$829$	−36.0118	−1.25074	−0.625370	−	0.780328i	$-0.715052\pi$
$829$	−36.0118	−1.25074	−0.625370	+	0.780328i	$0.715052\pi$
$830$	3.26682	0.113393
$831$	0	0
$832$	10.1404	0.351556
$833$	−4.08348	−0.141484
$834$	0	0
$835$	−15.2589	−0.528056
$836$	5.02730	0.173873
$837$	0	0
$838$	−10.4627	−0.361427
$839$	−34.2082	−1.18100	−0.590499	−	0.807038i	$-0.701069\pi$
$839$	−34.2082	−1.18100	−0.590499	+	0.807038i	$0.701069\pi$
$840$	0	0
$841$	72.1298	2.48723
$842$	−4.53253	−0.156201
$843$	0	0
$844$	10.2001	0.351102
$845$	18.9748	0.652752
$846$	0	0
$847$	−4.19208	−0.144041
$848$	−0.850419	−0.0292035
$849$	0	0
$850$	0.905100	0.0310447
$851$	−3.08859	−0.105875
$852$	0	0
$853$	11.3190	0.387554	0.193777	−	0.981046i	$-0.437926\pi$
$853$	11.3190	0.387554	0.193777	+	0.981046i	$0.437926\pi$
$854$	−2.29012	−0.0783664
$855$	0	0
$856$	5.74935	0.196509
$857$	−43.3860	−1.48204	−0.741019	−	0.671485i	$-0.765657\pi$
$857$	−43.3860	−1.48204	−0.741019	+	0.671485i	$0.765657\pi$
$858$	0	0
$859$	−25.6339	−0.874617	−0.437308	−	0.899312i	$-0.644068\pi$
$859$	−25.6339	−0.874617	−0.437308	+	0.899312i	$0.644068\pi$
$860$	7.16476	0.244316
$861$	0	0
$862$	9.66842	0.329308
$863$	−16.0782	−0.547309	−0.273654	−	0.961828i	$-0.588232\pi$
$863$	−16.0782	−0.547309	−0.273654	+	0.961828i	$0.588232\pi$
$864$	0	0
$865$	8.29759	0.282126
$866$	3.04789	0.103572
$867$	0	0
$868$	−75.8173	−2.57341
$869$	−8.67552	−0.294297
$870$	0	0
$871$	−58.4592	−1.98081
$872$	−39.3296	−1.33187
$873$	0	0
$874$	−1.57369	−0.0532308
$875$	32.8156	1.10937
$876$	0	0
$877$	−17.4124	−0.587974	−0.293987	−	0.955809i	$-0.594982\pi$
$877$	−17.4124	−0.587974	−0.293987	+	0.955809i	$0.594982\pi$
$878$	−3.77286	−0.127328
$879$	0	0
$880$	1.93672	0.0652867
$881$	41.4135	1.39526	0.697628	−	0.716460i	$-0.254239\pi$
$881$	41.4135	1.39526	0.697628	+	0.716460i	$0.254239\pi$
$882$	0	0
$883$	−49.5096	−1.66613	−0.833066	−	0.553174i	$-0.813416\pi$
$883$	−49.5096	−1.66613	−0.833066	+	0.553174i	$0.813416\pi$
$884$	−3.91639	−0.131722
$885$	0	0
$886$	−4.67038	−0.156905
$887$	48.4377	1.62638	0.813190	−	0.581999i	$-0.197729\pi$
$887$	48.4377	1.62638	0.813190	+	0.581999i	$0.197729\pi$
$888$	0	0
$889$	−4.98816	−0.167298
$890$	3.41762	0.114559
$891$	0	0
$892$	−2.05452	−0.0687905
$893$	23.2602	0.778372
$894$	0	0
$895$	8.45240	0.282533
$896$	48.3320	1.61466
$897$	0	0
$898$	12.0747	0.402938
$899$	−106.889	−3.56494
$900$	0	0
$901$	0.142895	0.00476052
$902$	0.706457	0.0235224
$903$	0	0
$904$	16.3583	0.544068
$905$	−11.1478	−0.370566
$906$	0	0
$907$	50.1623	1.66561	0.832807	−	0.553564i	$-0.186733\pi$
$907$	50.1623	1.66561	0.832807	+	0.553564i	$0.186733\pi$
$908$	5.39960	0.179192
$909$	0	0
$910$	11.5007	0.381244
$911$	19.2889	0.639071	0.319536	−	0.947574i	$-0.396473\pi$
$911$	19.2889	0.639071	0.319536	+	0.947574i	$0.396473\pi$
$912$	0	0
$913$	7.09673	0.234867
$914$	−6.69173	−0.221343
$915$	0	0
$916$	44.5509	1.47200
$917$	−0.973717	−0.0321550
$918$	0	0
$919$	−43.7882	−1.44444	−0.722220	−	0.691663i	$-0.756878\pi$
$919$	−43.7882	−1.44444	−0.722220	+	0.691663i	$0.756878\pi$
$920$	−1.66132	−0.0547720
$921$	0	0
$922$	3.01389	0.0992572
$923$	−26.0265	−0.856673
$924$	0	0
$925$	13.5898	0.446830
$926$	−21.3395	−0.701259
$927$	0	0
$928$	53.2979	1.74959
$929$	27.0615	0.887858	0.443929	−	0.896062i	$-0.353584\pi$
$929$	27.0615	0.887858	0.443929	+	0.896062i	$0.353584\pi$
$930$	0	0
$931$	−31.2397	−1.02384
$932$	26.1335	0.856032
$933$	0	0
$934$	8.44218	0.276237
$935$	−0.325424	−0.0106425
$936$	0	0
$937$	2.66965	0.0872137	0.0436068	−	0.999049i	$-0.486115\pi$
$937$	2.66965	0.0872137	0.0436068	+	0.999049i	$0.486115\pi$
$938$	−22.4639	−0.733471
$939$	0	0
$940$	11.2878	0.368167
$941$	31.2522	1.01879	0.509397	−	0.860532i	$-0.329868\pi$
$941$	31.2522	1.01879	0.509397	+	0.860532i	$0.329868\pi$
$942$	0	0
$943$	1.26083	0.0410583
$944$	−31.6544	−1.03026
$945$	0	0
$946$	−2.72990	−0.0887568
$947$	6.67071	0.216769	0.108384	−	0.994109i	$-0.465432\pi$
$947$	6.67071	0.216769	0.108384	+	0.994109i	$0.465432\pi$
$948$	0	0
$949$	12.3079	0.399532
$950$	6.92424	0.224652
$951$	0	0
$952$	−3.27383	−0.106105
$953$	−14.3407	−0.464540	−0.232270	−	0.972651i	$-0.574615\pi$
$953$	−14.3407	−0.464540	−0.232270	+	0.972651i	$0.574615\pi$
$954$	0	0
$955$	21.9648	0.710765
$956$	−11.0315	−0.356785
$957$	0	0
$958$	−10.4989	−0.339205
$959$	−57.8280	−1.86736
$960$	0	0
$961$	81.9756	2.64438
$962$	10.3137	0.332528
$963$	0	0
$964$	3.48848	0.112356
$965$	−1.39014	−0.0447501
$966$	0	0
$967$	33.8580	1.08880	0.544400	−	0.838826i	$-0.316757\pi$
$967$	33.8580	1.08880	0.544400	+	0.838826i	$0.316757\pi$
$968$	−2.02216	−0.0649945
$969$	0	0
$970$	−2.76629	−0.0888203
$971$	−23.2171	−0.745074	−0.372537	−	0.928017i	$-0.621512\pi$
$971$	−23.2171	−0.745074	−0.372537	+	0.928017i	$0.621512\pi$
$972$	0	0
$973$	−42.9383	−1.37654
$974$	−2.93180	−0.0939411
$975$	0	0
$976$	2.29842	0.0735705
$977$	−11.4832	−0.367381	−0.183690	−	0.982984i	$-0.558804\pi$
$977$	−11.4832	−0.367381	−0.183690	+	0.982984i	$0.558804\pi$
$978$	0	0
$979$	7.42432	0.237282
$980$	−15.1601	−0.484272
$981$	0	0
$982$	16.6067	0.529942
$983$	2.32576	0.0741803	0.0370901	−	0.999312i	$-0.488191\pi$
$983$	2.32576	0.0741803	0.0370901	+	0.999312i	$0.488191\pi$
$984$	0	0
$985$	12.6472	0.402974
$986$	−2.12169	−0.0675683
$987$	0	0
$988$	−29.9614	−0.953199
$989$	−4.87213	−0.154925
$990$	0	0
$991$	−27.4890	−0.873218	−0.436609	−	0.899651i	$-0.643821\pi$
$991$	−27.4890	−0.873218	−0.436609	+	0.899651i	$0.643821\pi$
$992$	−56.3330	−1.78857
$993$	0	0
$994$	−10.0011	−0.317216
$995$	0.364629	0.0115595
$996$	0	0
$997$	−0.204845	−0.00648751	−0.00324375	−	0.999995i	$-0.501033\pi$
$997$	−0.204845	−0.00648751	−0.00324375	+	0.999995i	$0.501033\pi$
$998$	−9.33754	−0.295575
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.9		14
3.2	odd	2		2013.2.a.h.1.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.6	✓	14	3.2	odd	2
6039.2.a.j.1.9		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.546298 q^{2} -1.70156 q^{4} +0.842631 q^{5} -4.19208 q^{7} -2.02216 q^{8} +O(q^{10})\)	\(q+0.546298 q^{2} -1.70156 q^{4} +0.842631 q^{5} -4.19208 q^{7} -2.02216 q^{8} +0.460328 q^{10} +1.00000 q^{11} -5.95974 q^{13} -2.29012 q^{14} +2.29842 q^{16} -0.386200 q^{17} -2.95453 q^{19} -1.43379 q^{20} +0.546298 q^{22} +0.974992 q^{23} -4.28997 q^{25} -3.25580 q^{26} +7.13306 q^{28} +10.0563 q^{29} -10.6290 q^{31} +5.29993 q^{32} -0.210980 q^{34} -3.53237 q^{35} -3.16781 q^{37} -1.61405 q^{38} -1.70393 q^{40} +1.29317 q^{41} -4.99709 q^{43} -1.70156 q^{44} +0.532636 q^{46} -7.87272 q^{47} +10.5735 q^{49} -2.34361 q^{50} +10.1408 q^{52} -0.370002 q^{53} +0.842631 q^{55} +8.47703 q^{56} +5.49376 q^{58} -13.7723 q^{59} +1.00000 q^{61} -5.80661 q^{62} -1.70149 q^{64} -5.02186 q^{65} +9.80901 q^{67} +0.657141 q^{68} -1.92973 q^{70} +4.36706 q^{71} -2.06518 q^{73} -1.73057 q^{74} +5.02730 q^{76} -4.19208 q^{77} -8.67552 q^{79} +1.93672 q^{80} +0.706457 q^{82} +7.09673 q^{83} -0.325424 q^{85} -2.72990 q^{86} -2.02216 q^{88} +7.42432 q^{89} +24.9837 q^{91} -1.65901 q^{92} -4.30086 q^{94} -2.48957 q^{95} -6.00940 q^{97} +5.77628 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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